
For \(X\)  proper, we obtain the result from Theorem REF . For \(X\)  qcqs and \(p \ne \ell \) , we can then choose a formal model of \(X\)  and deduce the claim from the constructibility of nearby cycles [1]}.

It is enough to prove that \(\mathrm {IC}_{X,\Lambda } = (\mathrm {IC}_{\mathcal {X},\Lambda })^{an}\) . This follows from the definition of either side as an appropriate image, and the compatibility of \((-)^{an}\)  with all the constituent operations (namely, \(j_!\) , \(j_*\) , perverse truncations, and images).

As \(X\)  is proper, Huber's results show that \(R\Gamma (X_C,-)\)  is compatible with duality [2]}, so it is enough to show that \(\mathbf {D}_X(IC_{X,\Lambda }) \simeq IC_{X,\Lambda }(d)\) . Using Theorem REF  (4), this amounts to checking that \(\mathbf {D}_U(\Lambda [d]) \simeq \Lambda [d](d) \)  for a smooth rigid space \(U\)  of dimension \(d\) , which follows immediately as \(\omega _U = \Lambda [2d](d)\)  (Remark REF ).

